Intruder state problem

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Intruder state problem is obviated by averaging of active orbital energies. 

It turns out that MR CISD represents again the most suitable source of the required higher-order clusters. Carefully chosen small reference space MR CISD involves a very small, yet representative, subset of such cluster amplitudes. Moreover, in this way we can also overcome the eventual intruder state problems by including such states in MR CISD, while excluding them from CMS SU CCSD. In other words, while we may have to exclude some references from Ado in order to avoid intruders, we can safely include them in the MR CISD model space Adi. In fact, we can even choose the CMS for Adi. Thus, designating the dimensions of Ado and Adi spaces by M and N, respectively, we refer to the ec SU CCSD method employing an NR-CISD as the external source by the acronym N, M)-CCSD. Thus, with this notation, we have that (N, 1)-CCSD = NR-RMR CCSD and (0, M)-CCSD = MR SU CCSD. Also, (0,1)-CCSD = SR CCSD. For details of this procedure and its applications we refer the reader to Refs. [63,64,71]. 

Hamiltonian has been diagonalized, a correction is applied for the appearance of the level shift in the denominators of the expressions for E,2). This level shift method has been applied successfully to a wide variety of problems in the field of spectroscopy and can be considered as a pragmatic solution to the intruder state problem inherent to perturbation theory. All CASSCF / CASPT2 calculations presented here have been done with MOLCAS-4 [27]. 

In spite of the intruder state problem, there is a range a systems where (relativistic) MBPT can produce reliable and accurate results for energies and width of resonance states. A few examples are given below. 

D. Nikolic, E. Lindroth, Intermediate Hamiltonian to avoid intruder state problems for doubly excited states, J. Phys. B 37 (2004) L285. 

The DC-CASCI (12,4) calculation was performed to construct reference functions. The active space includes the molecular spinors, which have atomic nature of 6p1/2 and 6p3/2 of Pb. This was followed with the DC-CASPT2 (24, 12, 160) level of calculation and the potential energy curves are illustrated in Figure 6-5a. This figure includes all the states which go to the first, second, and third dissociation channels, except 1U(II) state, which had intruder state problem. For simplification... 

Pad6 approximants to perturbation series have been considered by numerous authors,121 particularly those interested in effective interactions in nuclei. They have been used in studies of the intruder state problem which can arise when a multideterminantal function is used as a reference in a perturbative scheme.122... 

HF, while keeping the structure of the HF wave function as far as possible. In its fully variational form, it may yield a smooth potential surface with any kind of geometry, including extremely large bond distances, and it is free from the intruder-state problem (unfortunately, this does not hold for approximately optimized APSG). 

For avoiding the intruder state problem in BWCCSD we pay the price of losing size extensivity. Unfortunately, for the purpose of chemical application of MR BWCCSD, the size-extensivity error is not tolerable. For example, the MR BWCCSD value of the singlet-triplet gap in twisted ethylene exhibits an error of 3 kcal/mol with respect to full Cl [40]. Even worse case is the dissociation of the F2 molecule, where MR BWCCSD... 

There has been a revival of interest in the Brillouin-Wigner perturbation theory since it is seen as a possible remedy to the intruder state problem. As... 

Hubac and his co-workers222"231 have explored the use of Brillouin-Wigner perturbation theory in solving the coupled cluster equations. For the case of a single reference function, this approach is entirely equivalent to other formulations of the coupled cluster equations. However, for the multireference case, the Brillouin-Wigner coupled cluster theory shows some promise in that it appears to alleviate the intruder state problem. No doubt perturbative analysis will help to gain a deeper understanding of this approach. 

Some of the more important aspects of the theory behind the method are described in the review. In particular, the choice of the zeroth-order Hamiltonian is discussed together with the intruder-state problem and its solution. A generalization of the method to a multistate perturbation approach is suggested. Problems specifically related to spectroscopic applications are discussed, such as the choice of the active space and the treatment of solvent effects. 

The present review is concerned with general aspects of applications of the CASSCF/CASPT2 method in molecular spectroscopy. We first introduce the reader to the method. This is brief we refer readers to the original papers and the thesis of Andersson [14] for details. The pitfalls of the method are discussed with special emphasis on the intruder-state problem. The applications in spectroscopy are divided into two parts. One deals with recent results obtained for organic systems and the other with transition metal compounds. For each of these types of application, we discuss the general requirements that have to be fulfilled to obtain... 

State computed using the active space (66) due to intruder-state problems. 

In summary, there is agreement between the spectroscopic data computed and available experimental information. The CASSCF/ CASPT2 method has thus shown that it is capable of treating not only the ground state but also a large number of excited states for a molecule that has been a frustrating challenge for quantum chemistry for more than 10 years. With the newly implemented level-shift technique, the intruder state problem is also solved, which makes the calculations less involved and the results more stable. The entire field of spectroscopy of transition metal dimers and their ions is therefore open for an accurate theoretical analysis. 

---